(challenge) solve this without looking it up.

I have not looked it up, but I did read it a long while ago (about 2 years ago)

albert knows the month
bernard knows the day

first:
albert doesn't know, so his month that he is given must repeat at least once

bernard doesn't know at first, so the day must repeat as well, so it can't be the 19 nor the 16th

the only way for albert to know that bernard doesn't know is that albert must not have been told a month with 16 or 19 in it
that eliminates may and june, leaving july and august

2nd line, now bernard deduces that albert must have been told either july or august
he could not have been told 14 since he would not be able to tell if it was july or august
he was therefor told 15,16, or 17

3rd line:
now albert knows that it is either 15,16,17
notice that july is 16
15 and 17 are both on august
for albert to know what the date is, he must have been told july ( the only one with 1 date on it)

so the birthday is july 16